Class TTest
 java.lang.Object

 org.apache.commons.math4.legacy.stat.inference.TTest

public class TTest extends Object
An implementation for Student's ttests.Tests can be:
 Onesample or twosample
 Onesided or twosided
 Paired or unpaired (for twosample tests)
 Homoscedastic (equal variance assumption) or heteroscedastic (for two sample tests)
 Fixed significance level (booleanvalued) or returning pvalues.
Test statistics are available for all tests. Methods including "Test" in in their names perform tests, all other methods return tstatistics. Among the "Test" methods,
double
valued methods return pvalues;boolean
valued methods perform fixed significance level tests. Significance levels are always specified as numbers between 0 and 0.5 (e.g. tests at the 95% level usealpha=0.05
).Input to tests can be either
double[]
arrays orStatisticalSummary
instances.Uses commonsmath
TDistribution
implementation to estimate exact pvalues.


Constructor Summary
Constructors Constructor Description TTest()

Method Summary
All Methods Instance Methods Concrete Methods Modifier and Type Method Description protected double
df(double v1, double v2, double n1, double n2)
Computes approximate degrees of freedom for 2sample ttest.double
homoscedasticT(double[] sample1, double[] sample2)
Computes a 2sample t statistic, under the hypothesis of equal subpopulation variances.protected double
homoscedasticT(double m1, double m2, double v1, double v2, double n1, double n2)
Computes t test statistic for 2sample ttest under the hypothesis of equal subpopulation variances.double
homoscedasticT(StatisticalSummary sampleStats1, StatisticalSummary sampleStats2)
Computes a 2sample t statistic, comparing the means of the datasets described by twoStatisticalSummary
instances, under the assumption of equal subpopulation variances.double
homoscedasticTTest(double[] sample1, double[] sample2)
Returns the observed significance level, or pvalue, associated with a twosample, twotailed ttest comparing the means of the input arrays, under the assumption that the two samples are drawn from subpopulations with equal variances.boolean
homoscedasticTTest(double[] sample1, double[] sample2, double alpha)
Performs a twosided ttest evaluating the null hypothesis thatsample1
andsample2
are drawn from populations with the same mean, with significance levelalpha
, assuming that the subpopulation variances are equal.protected double
homoscedasticTTest(double m1, double m2, double v1, double v2, double n1, double n2)
Computes pvalue for 2sided, 2sample ttest, under the assumption of equal subpopulation variances.double
homoscedasticTTest(StatisticalSummary sampleStats1, StatisticalSummary sampleStats2)
Returns the observed significance level, or pvalue, associated with a twosample, twotailed ttest comparing the means of the datasets described by two StatisticalSummary instances, under the hypothesis of equal subpopulation variances.double
pairedT(double[] sample1, double[] sample2)
Computes a paired, 2sample tstatistic based on the data in the input arrays.double
pairedTTest(double[] sample1, double[] sample2)
Returns the observed significance level, or pvalue, associated with a paired, twosample, twotailed ttest based on the data in the input arrays.boolean
pairedTTest(double[] sample1, double[] sample2, double alpha)
Performs a paired ttest evaluating the null hypothesis that the mean of the paired differences betweensample1
andsample2
is 0 in favor of the twosided alternative that the mean paired difference is not equal to 0, with significance levelalpha
.double
t(double[] sample1, double[] sample2)
Computes a 2sample t statistic, without the hypothesis of equal subpopulation variances.double
t(double mu, double[] observed)
Computes a t statistic given observed values and a comparison constant.protected double
t(double m, double mu, double v, double n)
Computes t test statistic for 1sample ttest.protected double
t(double m1, double m2, double v1, double v2, double n1, double n2)
Computes t test statistic for 2sample ttest.double
t(double mu, StatisticalSummary sampleStats)
double
t(StatisticalSummary sampleStats1, StatisticalSummary sampleStats2)
Computes a 2sample t statistic, comparing the means of the datasets described by twoStatisticalSummary
instances, without the assumption of equal subpopulation variances.double
tTest(double[] sample1, double[] sample2)
Returns the observed significance level, or pvalue, associated with a twosample, twotailed ttest comparing the means of the input arrays.boolean
tTest(double[] sample1, double[] sample2, double alpha)
Performs a twosided ttest evaluating the null hypothesis thatsample1
andsample2
are drawn from populations with the same mean, with significance levelalpha
.double
tTest(double mu, double[] sample)
Returns the observed significance level, or pvalue, associated with a onesample, twotailed ttest comparing the mean of the input array with the constantmu
.boolean
tTest(double mu, double[] sample, double alpha)
Performs a twosided ttest evaluating the null hypothesis that the mean of the population from whichsample
is drawn equalsmu
.protected double
tTest(double m, double mu, double v, double n)
Computes pvalue for 2sided, 1sample ttest.protected double
tTest(double m1, double m2, double v1, double v2, double n1, double n2)
Computes pvalue for 2sided, 2sample ttest.double
tTest(double mu, StatisticalSummary sampleStats)
Returns the observed significance level, or pvalue, associated with a onesample, twotailed ttest comparing the mean of the dataset described bysampleStats
with the constantmu
.boolean
tTest(double mu, StatisticalSummary sampleStats, double alpha)
Performs a twosided ttest evaluating the null hypothesis that the mean of the population from which the dataset described bystats
is drawn equalsmu
.double
tTest(StatisticalSummary sampleStats1, StatisticalSummary sampleStats2)
Returns the observed significance level, or pvalue, associated with a twosample, twotailed ttest comparing the means of the datasets described by two StatisticalSummary instances.boolean
tTest(StatisticalSummary sampleStats1, StatisticalSummary sampleStats2, double alpha)
Performs a twosided ttest evaluating the null hypothesis thatsampleStats1
andsampleStats2
describe datasets drawn from populations with the same mean, with significance levelalpha
.



Constructor Detail

TTest
public TTest()


Method Detail

pairedT
public double pairedT(double[] sample1, double[] sample2) throws NullArgumentException, NoDataException, DimensionMismatchException, NumberIsTooSmallException
Computes a paired, 2sample tstatistic based on the data in the input arrays. The tstatistic returned is equivalent to what would be returned by computing the onesample tstatistict(double, double[])
, withmu = 0
and the sample array consisting of the (signed) differences between corresponding entries insample1
andsample2.
Preconditions:
 The input arrays must have the same length and their common length must be at least 2.
 Parameters:
sample1
 array of sample data valuessample2
 array of sample data values Returns:
 t statistic
 Throws:
NullArgumentException
 if the arrays arenull
NoDataException
 if the arrays are emptyDimensionMismatchException
 if the length of the arrays is not equalNumberIsTooSmallException
 if the length of the arrays is < 2

pairedTTest
public double pairedTTest(double[] sample1, double[] sample2) throws NullArgumentException, NoDataException, DimensionMismatchException, NumberIsTooSmallException, MaxCountExceededException
Returns the observed significance level, or pvalue, associated with a paired, twosample, twotailed ttest based on the data in the input arrays.The number returned is the smallest significance level at which one can reject the null hypothesis that the mean of the paired differences is 0 in favor of the twosided alternative that the mean paired difference is not equal to 0. For a onesided test, divide the returned value by 2.
This test is equivalent to a onesample ttest computed using
tTest(double, double[])
withmu = 0
and the sample array consisting of the signed differences between corresponding elements ofsample1
andsample2.
Usage Note:
The validity of the pvalue depends on the assumptions of the parametric ttest procedure, as discussed herePreconditions:
 The input array lengths must be the same and their common length must be at least 2.
 Parameters:
sample1
 array of sample data valuessample2
 array of sample data values Returns:
 pvalue for ttest
 Throws:
NullArgumentException
 if the arrays arenull
NoDataException
 if the arrays are emptyDimensionMismatchException
 if the length of the arrays is not equalNumberIsTooSmallException
 if the length of the arrays is < 2MaxCountExceededException
 if an error occurs computing the pvalue

pairedTTest
public boolean pairedTTest(double[] sample1, double[] sample2, double alpha) throws NullArgumentException, NoDataException, DimensionMismatchException, NumberIsTooSmallException, OutOfRangeException, MaxCountExceededException
Performs a paired ttest evaluating the null hypothesis that the mean of the paired differences betweensample1
andsample2
is 0 in favor of the twosided alternative that the mean paired difference is not equal to 0, with significance levelalpha
.Returns
true
iff the null hypothesis can be rejected with confidence1  alpha
. To perform a 1sided test, usealpha * 2
Usage Note:
The validity of the test depends on the assumptions of the parametric ttest procedure, as discussed herePreconditions:
 The input array lengths must be the same and their common length must be at least 2.

0 < alpha < 0.5
 Parameters:
sample1
 array of sample data valuessample2
 array of sample data valuesalpha
 significance level of the test Returns:
 true if the null hypothesis can be rejected with confidence 1  alpha
 Throws:
NullArgumentException
 if the arrays arenull
NoDataException
 if the arrays are emptyDimensionMismatchException
 if the length of the arrays is not equalNumberIsTooSmallException
 if the length of the arrays is < 2OutOfRangeException
 ifalpha
is not in the range (0, 0.5]MaxCountExceededException
 if an error occurs computing the pvalue

t
public double t(double mu, double[] observed) throws NullArgumentException, NumberIsTooSmallException
Computes a t statistic given observed values and a comparison constant.This statistic can be used to perform a one sample ttest for the mean.
Preconditions:
 The observed array length must be at least 2.
 Parameters:
mu
 comparison constantobserved
 array of values Returns:
 t statistic
 Throws:
NullArgumentException
 ifobserved
isnull
NumberIsTooSmallException
 if the length ofobserved
is < 2

t
public double t(double mu, StatisticalSummary sampleStats) throws NullArgumentException, NumberIsTooSmallException
Computes a t statistic to use in comparing the mean of the dataset described bysampleStats
tomu
.This statistic can be used to perform a one sample ttest for the mean.
Preconditions:
observed.getN() ≥ 2
.
 Parameters:
mu
 comparison constantsampleStats
 DescriptiveStatistics holding sample summary statitstics Returns:
 t statistic
 Throws:
NullArgumentException
 ifsampleStats
isnull
NumberIsTooSmallException
 if the number of samples is < 2

homoscedasticT
public double homoscedasticT(double[] sample1, double[] sample2) throws NullArgumentException, NumberIsTooSmallException
Computes a 2sample t statistic, under the hypothesis of equal subpopulation variances. To compute a tstatistic without the equal variances hypothesis, uset(double[], double[])
.This statistic can be used to perform a (homoscedastic) twosample ttest to compare sample means.
The tstatistic is
t = (m1  m2) / (sqrt(1/n1 +1/n2) sqrt(var))
where
n1
is the size of first sample;n2
is the size of second sample;m1
is the mean of first sample;m2
is the mean of second sample andvar
is the pooled variance estimate:var = sqrt(((n1  1)var1 + (n2  1)var2) / ((n11) + (n21)))
with
var1
the variance of the first sample andvar2
the variance of the second sample.Preconditions:
 The observed array lengths must both be at least 2.
 Parameters:
sample1
 array of sample data valuessample2
 array of sample data values Returns:
 t statistic
 Throws:
NullArgumentException
 if the arrays arenull
NumberIsTooSmallException
 if the length of the arrays is < 2

t
public double t(double[] sample1, double[] sample2) throws NullArgumentException, NumberIsTooSmallException
Computes a 2sample t statistic, without the hypothesis of equal subpopulation variances. To compute a tstatistic assuming equal variances, usehomoscedasticT(double[], double[])
.This statistic can be used to perform a twosample ttest to compare sample means.
The tstatistic is
t = (m1  m2) / sqrt(var1/n1 + var2/n2)
where
n1
is the size of the first samplen2
is the size of the second sample;m1
is the mean of the first sample;m2
is the mean of the second sample;var1
is the variance of the first sample;var2
is the variance of the second sample;Preconditions:
 The observed array lengths must both be at least 2.
 Parameters:
sample1
 array of sample data valuessample2
 array of sample data values Returns:
 t statistic
 Throws:
NullArgumentException
 if the arrays arenull
NumberIsTooSmallException
 if the length of the arrays is < 2

t
public double t(StatisticalSummary sampleStats1, StatisticalSummary sampleStats2) throws NullArgumentException, NumberIsTooSmallException
Computes a 2sample t statistic, comparing the means of the datasets described by twoStatisticalSummary
instances, without the assumption of equal subpopulation variances. UsehomoscedasticT(StatisticalSummary, StatisticalSummary)
to compute a tstatistic under the equal variances assumption.This statistic can be used to perform a twosample ttest to compare sample means.
The returned tstatistic is
t = (m1  m2) / sqrt(var1/n1 + var2/n2)
where
n1
is the size of the first sample;n2
is the size of the second sample;m1
is the mean of the first sample;m2
is the mean of the second samplevar1
is the variance of the first sample;var2
is the variance of the second samplePreconditions:
 The datasets described by the two Univariates must each contain at least 2 observations.
 Parameters:
sampleStats1
 StatisticalSummary describing data from the first samplesampleStats2
 StatisticalSummary describing data from the second sample Returns:
 t statistic
 Throws:
NullArgumentException
 if the sample statistics arenull
NumberIsTooSmallException
 if the number of samples is < 2

homoscedasticT
public double homoscedasticT(StatisticalSummary sampleStats1, StatisticalSummary sampleStats2) throws NullArgumentException, NumberIsTooSmallException
Computes a 2sample t statistic, comparing the means of the datasets described by twoStatisticalSummary
instances, under the assumption of equal subpopulation variances. To compute a tstatistic without the equal variances assumption, uset(StatisticalSummary, StatisticalSummary)
.This statistic can be used to perform a (homoscedastic) twosample ttest to compare sample means.
The tstatistic returned is
t = (m1  m2) / (sqrt(1/n1 +1/n2) sqrt(var))
where
n1
is the size of first sample;n2
is the size of second sample;m1
is the mean of first sample;m2
is the mean of second sample andvar
is the pooled variance estimate:var = sqrt(((n1  1)var1 + (n2  1)var2) / ((n11) + (n21)))
with
var1
the variance of the first sample andvar2
the variance of the second sample.Preconditions:
 The datasets described by the two Univariates must each contain at least 2 observations.
 Parameters:
sampleStats1
 StatisticalSummary describing data from the first samplesampleStats2
 StatisticalSummary describing data from the second sample Returns:
 t statistic
 Throws:
NullArgumentException
 if the sample statistics arenull
NumberIsTooSmallException
 if the number of samples is < 2

tTest
public double tTest(double mu, double[] sample) throws NullArgumentException, NumberIsTooSmallException, MaxCountExceededException
Returns the observed significance level, or pvalue, associated with a onesample, twotailed ttest comparing the mean of the input array with the constantmu
.The number returned is the smallest significance level at which one can reject the null hypothesis that the mean equals
mu
in favor of the twosided alternative that the mean is different frommu
. For a onesided test, divide the returned value by 2.Usage Note:
The validity of the test depends on the assumptions of the parametric ttest procedure, as discussed herePreconditions:
 The observed array length must be at least 2.
 Parameters:
mu
 constant value to compare sample mean againstsample
 array of sample data values Returns:
 pvalue
 Throws:
NullArgumentException
 if the sample array isnull
NumberIsTooSmallException
 if the length of the array is < 2MaxCountExceededException
 if an error occurs computing the pvalue

tTest
public boolean tTest(double mu, double[] sample, double alpha) throws NullArgumentException, NumberIsTooSmallException, OutOfRangeException, MaxCountExceededException
Performs a twosided ttest evaluating the null hypothesis that the mean of the population from whichsample
is drawn equalsmu
.Returns
true
iff the null hypothesis can be rejected with confidence1  alpha
. To perform a 1sided test, usealpha * 2
Examples:
 To test the (2sided) hypothesis
sample mean = mu
at the 95% level, usetTest(mu, sample, 0.05)
 To test the (onesided) hypothesis
sample mean < mu
at the 99% level, first verify that the measured sample mean is less thanmu
and then usetTest(mu, sample, 0.02)
Usage Note:
The validity of the test depends on the assumptions of the onesample parametric ttest procedure, as discussed herePreconditions:
 The observed array length must be at least 2.
 Parameters:
mu
 constant value to compare sample mean againstsample
 array of sample data valuesalpha
 significance level of the test Returns:
 pvalue
 Throws:
NullArgumentException
 if the sample array isnull
NumberIsTooSmallException
 if the length of the array is < 2OutOfRangeException
 ifalpha
is not in the range (0, 0.5]MaxCountExceededException
 if an error computing the pvalue
 To test the (2sided) hypothesis

tTest
public double tTest(double mu, StatisticalSummary sampleStats) throws NullArgumentException, NumberIsTooSmallException, MaxCountExceededException
Returns the observed significance level, or pvalue, associated with a onesample, twotailed ttest comparing the mean of the dataset described bysampleStats
with the constantmu
.The number returned is the smallest significance level at which one can reject the null hypothesis that the mean equals
mu
in favor of the twosided alternative that the mean is different frommu
. For a onesided test, divide the returned value by 2.Usage Note:
The validity of the test depends on the assumptions of the parametric ttest procedure, as discussed herePreconditions:
 The sample must contain at least 2 observations.
 Parameters:
mu
 constant value to compare sample mean againstsampleStats
 StatisticalSummary describing sample data Returns:
 pvalue
 Throws:
NullArgumentException
 ifsampleStats
isnull
NumberIsTooSmallException
 if the number of samples is < 2MaxCountExceededException
 if an error occurs computing the pvalue

tTest
public boolean tTest(double mu, StatisticalSummary sampleStats, double alpha) throws NullArgumentException, NumberIsTooSmallException, OutOfRangeException, MaxCountExceededException
Performs a twosided ttest evaluating the null hypothesis that the mean of the population from which the dataset described bystats
is drawn equalsmu
.Returns
true
iff the null hypothesis can be rejected with confidence1  alpha
. To perform a 1sided test, usealpha * 2.
Examples:
 To test the (2sided) hypothesis
sample mean = mu
at the 95% level, usetTest(mu, sampleStats, 0.05)
 To test the (onesided) hypothesis
sample mean < mu
at the 99% level, first verify that the measured sample mean is less thanmu
and then usetTest(mu, sampleStats, 0.02)
Usage Note:
The validity of the test depends on the assumptions of the onesample parametric ttest procedure, as discussed herePreconditions:
 The sample must include at least 2 observations.
 Parameters:
mu
 constant value to compare sample mean againstsampleStats
 StatisticalSummary describing sample data valuesalpha
 significance level of the test Returns:
 pvalue
 Throws:
NullArgumentException
 ifsampleStats
isnull
NumberIsTooSmallException
 if the number of samples is < 2OutOfRangeException
 ifalpha
is not in the range (0, 0.5]MaxCountExceededException
 if an error occurs computing the pvalue
 To test the (2sided) hypothesis

tTest
public double tTest(double[] sample1, double[] sample2) throws NullArgumentException, NumberIsTooSmallException, MaxCountExceededException
Returns the observed significance level, or pvalue, associated with a twosample, twotailed ttest comparing the means of the input arrays.The number returned is the smallest significance level at which one can reject the null hypothesis that the two means are equal in favor of the twosided alternative that they are different. For a onesided test, divide the returned value by 2.
The test does not assume that the underlying popuation variances are equal and it uses approximated degrees of freedom computed from the sample data to compute the pvalue. The tstatistic used is as defined in
t(double[], double[])
and the WelchSatterthwaite approximation to the degrees of freedom is used, as described here. To perform the test under the assumption of equal subpopulation variances, usehomoscedasticTTest(double[], double[])
.Usage Note:
The validity of the pvalue depends on the assumptions of the parametric ttest procedure, as discussed herePreconditions:
 The observed array lengths must both be at least 2.
 Parameters:
sample1
 array of sample data valuessample2
 array of sample data values Returns:
 pvalue for ttest
 Throws:
NullArgumentException
 if the arrays arenull
NumberIsTooSmallException
 if the length of the arrays is < 2MaxCountExceededException
 if an error occurs computing the pvalue

homoscedasticTTest
public double homoscedasticTTest(double[] sample1, double[] sample2) throws NullArgumentException, NumberIsTooSmallException, MaxCountExceededException
Returns the observed significance level, or pvalue, associated with a twosample, twotailed ttest comparing the means of the input arrays, under the assumption that the two samples are drawn from subpopulations with equal variances. To perform the test without the equal variances assumption, usetTest(double[], double[])
.The number returned is the smallest significance level at which one can reject the null hypothesis that the two means are equal in favor of the twosided alternative that they are different. For a onesided test, divide the returned value by 2.
A pooled variance estimate is used to compute the tstatistic. See
homoscedasticT(double[], double[])
. The sum of the sample sizes minus 2 is used as the degrees of freedom.Usage Note:
The validity of the pvalue depends on the assumptions of the parametric ttest procedure, as discussed herePreconditions:
 The observed array lengths must both be at least 2.
 Parameters:
sample1
 array of sample data valuessample2
 array of sample data values Returns:
 pvalue for ttest
 Throws:
NullArgumentException
 if the arrays arenull
NumberIsTooSmallException
 if the length of the arrays is < 2MaxCountExceededException
 if an error occurs computing the pvalue

tTest
public boolean tTest(double[] sample1, double[] sample2, double alpha) throws NullArgumentException, NumberIsTooSmallException, OutOfRangeException, MaxCountExceededException
Performs a twosided ttest evaluating the null hypothesis thatsample1
andsample2
are drawn from populations with the same mean, with significance levelalpha
. This test does not assume that the subpopulation variances are equal. To perform the test assuming equal variances, usehomoscedasticTTest(double[], double[], double)
.Returns
true
iff the null hypothesis that the means are equal can be rejected with confidence1  alpha
. To perform a 1sided test, usealpha * 2
See
t(double[], double[])
for the formula used to compute the tstatistic. Degrees of freedom are approximated using the WelchSatterthwaite approximation.Examples:
 To test the (2sided) hypothesis
mean 1 = mean 2
at the 95% level, usetTest(sample1, sample2, 0.05).
 To test the (onesided) hypothesis
mean 1 < mean 2
, at the 99% level, first verify that the measured mean ofsample 1
is less than the mean ofsample 2
and then usetTest(sample1, sample2, 0.02)
Usage Note:
The validity of the test depends on the assumptions of the parametric ttest procedure, as discussed herePreconditions:
 The observed array lengths must both be at least 2.

0 < alpha < 0.5
 Parameters:
sample1
 array of sample data valuessample2
 array of sample data valuesalpha
 significance level of the test Returns:
 true if the null hypothesis can be rejected with confidence 1  alpha
 Throws:
NullArgumentException
 if the arrays arenull
NumberIsTooSmallException
 if the length of the arrays is < 2OutOfRangeException
 ifalpha
is not in the range (0, 0.5]MaxCountExceededException
 if an error occurs computing the pvalue
 To test the (2sided) hypothesis

homoscedasticTTest
public boolean homoscedasticTTest(double[] sample1, double[] sample2, double alpha) throws NullArgumentException, NumberIsTooSmallException, OutOfRangeException, MaxCountExceededException
Performs a twosided ttest evaluating the null hypothesis thatsample1
andsample2
are drawn from populations with the same mean, with significance levelalpha
, assuming that the subpopulation variances are equal. UsetTest(double[], double[], double)
to perform the test without the assumption of equal variances.Returns
true
iff the null hypothesis that the means are equal can be rejected with confidence1  alpha
. To perform a 1sided test, usealpha * 2.
To perform the test without the assumption of equal subpopulation variances, usetTest(double[], double[], double)
.A pooled variance estimate is used to compute the tstatistic. See
t(double[], double[])
for the formula. The sum of the sample sizes minus 2 is used as the degrees of freedom.Examples:
 To test the (2sided) hypothesis
mean 1 = mean 2
at the 95% level, usetTest(sample1, sample2, 0.05).
 To test the (onesided) hypothesis
mean 1 < mean 2,
at the 99% level, first verify that the measured mean ofsample 1
is less than the mean ofsample 2
and then usetTest(sample1, sample2, 0.02)
Usage Note:
The validity of the test depends on the assumptions of the parametric ttest procedure, as discussed herePreconditions:
 The observed array lengths must both be at least 2.

0 < alpha < 0.5
 Parameters:
sample1
 array of sample data valuessample2
 array of sample data valuesalpha
 significance level of the test Returns:
 true if the null hypothesis can be rejected with confidence 1  alpha
 Throws:
NullArgumentException
 if the arrays arenull
NumberIsTooSmallException
 if the length of the arrays is < 2OutOfRangeException
 ifalpha
is not in the range (0, 0.5]MaxCountExceededException
 if an error occurs computing the pvalue
 To test the (2sided) hypothesis

tTest
public double tTest(StatisticalSummary sampleStats1, StatisticalSummary sampleStats2) throws NullArgumentException, NumberIsTooSmallException, MaxCountExceededException
Returns the observed significance level, or pvalue, associated with a twosample, twotailed ttest comparing the means of the datasets described by two StatisticalSummary instances.The number returned is the smallest significance level at which one can reject the null hypothesis that the two means are equal in favor of the twosided alternative that they are different. For a onesided test, divide the returned value by 2.
The test does not assume that the underlying population variances are equal and it uses approximated degrees of freedom computed from the sample data to compute the pvalue. To perform the test assuming equal variances, use
homoscedasticTTest(StatisticalSummary, StatisticalSummary)
.Usage Note:
The validity of the pvalue depends on the assumptions of the parametric ttest procedure, as discussed herePreconditions:
 The datasets described by the two Univariates must each contain at least 2 observations.
 Parameters:
sampleStats1
 StatisticalSummary describing data from the first samplesampleStats2
 StatisticalSummary describing data from the second sample Returns:
 pvalue for ttest
 Throws:
NullArgumentException
 if the sample statistics arenull
NumberIsTooSmallException
 if the number of samples is < 2MaxCountExceededException
 if an error occurs computing the pvalue

homoscedasticTTest
public double homoscedasticTTest(StatisticalSummary sampleStats1, StatisticalSummary sampleStats2) throws NullArgumentException, NumberIsTooSmallException, MaxCountExceededException
Returns the observed significance level, or pvalue, associated with a twosample, twotailed ttest comparing the means of the datasets described by two StatisticalSummary instances, under the hypothesis of equal subpopulation variances. To perform a test without the equal variances assumption, usetTest(StatisticalSummary, StatisticalSummary)
.The number returned is the smallest significance level at which one can reject the null hypothesis that the two means are equal in favor of the twosided alternative that they are different. For a onesided test, divide the returned value by 2.
See
homoscedasticT(double[], double[])
for the formula used to compute the tstatistic. The sum of the sample sizes minus 2 is used as the degrees of freedom.Usage Note:
The validity of the pvalue depends on the assumptions of the parametric ttest procedure, as discussed herePreconditions:
 The datasets described by the two Univariates must each contain at least 2 observations.
 Parameters:
sampleStats1
 StatisticalSummary describing data from the first samplesampleStats2
 StatisticalSummary describing data from the second sample Returns:
 pvalue for ttest
 Throws:
NullArgumentException
 if the sample statistics arenull
NumberIsTooSmallException
 if the number of samples is < 2MaxCountExceededException
 if an error occurs computing the pvalue

tTest
public boolean tTest(StatisticalSummary sampleStats1, StatisticalSummary sampleStats2, double alpha) throws NullArgumentException, NumberIsTooSmallException, OutOfRangeException, MaxCountExceededException
Performs a twosided ttest evaluating the null hypothesis thatsampleStats1
andsampleStats2
describe datasets drawn from populations with the same mean, with significance levelalpha
. This test does not assume that the subpopulation variances are equal. To perform the test under the equal variances assumption, usehomoscedasticTTest(StatisticalSummary, StatisticalSummary)
.Returns
true
iff the null hypothesis that the means are equal can be rejected with confidence1  alpha
. To perform a 1sided test, usealpha * 2
See
t(double[], double[])
for the formula used to compute the tstatistic. Degrees of freedom are approximated using the WelchSatterthwaite approximation.Examples:
 To test the (2sided) hypothesis
mean 1 = mean 2
at the 95%, usetTest(sampleStats1, sampleStats2, 0.05)
 To test the (onesided) hypothesis
mean 1 < mean 2
at the 99% level, first verify that the measured mean ofsample 1
is less than the mean ofsample 2
and then usetTest(sampleStats1, sampleStats2, 0.02)
Usage Note:
The validity of the test depends on the assumptions of the parametric ttest procedure, as discussed herePreconditions:
 The datasets described by the two Univariates must each contain at least 2 observations.

0 < alpha < 0.5
 Parameters:
sampleStats1
 StatisticalSummary describing sample data valuessampleStats2
 StatisticalSummary describing sample data valuesalpha
 significance level of the test Returns:
 true if the null hypothesis can be rejected with confidence 1  alpha
 Throws:
NullArgumentException
 if the sample statistics arenull
NumberIsTooSmallException
 if the number of samples is < 2OutOfRangeException
 ifalpha
is not in the range (0, 0.5]MaxCountExceededException
 if an error occurs computing the pvalue
 To test the (2sided) hypothesis

df
protected double df(double v1, double v2, double n1, double n2)
Computes approximate degrees of freedom for 2sample ttest. Parameters:
v1
 first sample variancev2
 second sample variancen1
 first sample nn2
 second sample n Returns:
 approximate degrees of freedom

t
protected double t(double m, double mu, double v, double n)
Computes t test statistic for 1sample ttest. Parameters:
m
 sample meanmu
 constant to test againstv
 sample variancen
 sample n Returns:
 t test statistic

t
protected double t(double m1, double m2, double v1, double v2, double n1, double n2)
Computes t test statistic for 2sample ttest.Does not assume that subpopulation variances are equal.
 Parameters:
m1
 first sample meanm2
 second sample meanv1
 first sample variancev2
 second sample variancen1
 first sample nn2
 second sample n Returns:
 t test statistic

homoscedasticT
protected double homoscedasticT(double m1, double m2, double v1, double v2, double n1, double n2)
Computes t test statistic for 2sample ttest under the hypothesis of equal subpopulation variances. Parameters:
m1
 first sample meanm2
 second sample meanv1
 first sample variancev2
 second sample variancen1
 first sample nn2
 second sample n Returns:
 t test statistic

tTest
protected double tTest(double m, double mu, double v, double n) throws MaxCountExceededException, MathIllegalArgumentException
Computes pvalue for 2sided, 1sample ttest. Parameters:
m
 sample meanmu
 constant to test againstv
 sample variancen
 sample n Returns:
 pvalue
 Throws:
MaxCountExceededException
 if an error occurs computing the pvalueMathIllegalArgumentException
 if n is not greater than 1

tTest
protected double tTest(double m1, double m2, double v1, double v2, double n1, double n2) throws MaxCountExceededException, NotStrictlyPositiveException
Computes pvalue for 2sided, 2sample ttest.Does not assume subpopulation variances are equal. Degrees of freedom are estimated from the data.
 Parameters:
m1
 first sample meanm2
 second sample meanv1
 first sample variancev2
 second sample variancen1
 first sample nn2
 second sample n Returns:
 pvalue
 Throws:
MaxCountExceededException
 if an error occurs computing the pvalueNotStrictlyPositiveException
 if the estimated degrees of freedom is not strictly positive

homoscedasticTTest
protected double homoscedasticTTest(double m1, double m2, double v1, double v2, double n1, double n2) throws MaxCountExceededException, NotStrictlyPositiveException
Computes pvalue for 2sided, 2sample ttest, under the assumption of equal subpopulation variances.The sum of the sample sizes minus 2 is used as degrees of freedom.
 Parameters:
m1
 first sample meanm2
 second sample meanv1
 first sample variancev2
 second sample variancen1
 first sample nn2
 second sample n Returns:
 pvalue
 Throws:
MaxCountExceededException
 if an error occurs computing the pvalueNotStrictlyPositiveException
 if the estimated degrees of freedom is not strictly positive

